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New York Journal of Mathematics a Note on Lattices of Z-Ideals of F-Rings New York Journal of Mathematics New York J. Math. 22 (2016) 351{361. A note on lattices of z-ideals of f-rings Themba Dube Abstract. The lattice of z-ideals of the ring C(X) of real-valued con- tinuous functions on a completely regular Hausdorff space X has been shown by Mart´ınezand Zenk to be a complete Heyting algebra with cer- tain properties. We show that these properties are due only to the fact that C(X) is an f-ring with bounded inversion. This we do by studying lattices of algebraic z-ideals of abstract f-rings with bounded inversion. Contents 1. Introduction 351 2. Preliminaries 352 2.1. Rings and f-rings 352 2.2. Algebraic frames 352 3. Frames of z-ideals of f-rings 353 4. Functoriality emanating from z-ideals 359 References 361 1. Introduction It has been remarked by Subramanian [13] that many results in the ring C(X) of real-valued continuous functions on a completely regular Hausdorff space X are mainly due to the fact that C(X) is an f-ring. Of course C(X) is not just any f-ring; it also has some special properties such as bounded inversion, which means that every element above the identity (relative to its natural partial order) is invertible. It also has the property that the sum of two z-ideals in it is a z-ideal [3]. In the course of this introduction we shall use terms which (although standard) we will recall, but only in subsequent sections. Our aim in this paper is to show that it is because C(X) is an f-ring with bounded inversion that the lattice Cz(X) of z-ideals of C(X) is a normal Received January 26, 2016. 2010 Mathematics Subject Classification. Primary: 06F25; Secondary: 06D22, 13A15, 18A05. Key words and phrases. f-ring, z-ideal, coherent frame, proper map, functor, natural transformation. The author acknowledges funding from the National Research Foundation of South Africa through a research grant with Grant No. 93514. ISSN 1076-9803/2016 351 352 THEMBA DUBE coherent Yosida frame. This we do by showing that if A is any f-ring with bounded inversion, then the lattice ZId(A) of z-ideals of A (here we mean z-ideals in the algebraic sense) is a normal coherent Yosida frame (Theorems 3.2 and 3.5). We hasten to mention that an ideal of C(X) is a z-ideal algebraically if and only if it is z-ideal in the usual topological sense. We show that ZId(A) is a frame by constructing a nucleus, z: RL(A) ! RL(A); on the frame RL(A) of radical `-ideals of A whose image is precisely ZId(A). As it turns out, this nucleus induces a codense coherent map RL(A) ! ZId(A); which is dense precisely when the Jacobson radical and the nilradical of A coincide (Proposition 3.10). Furthermore, it leads to a natural transforma- tion RL ! ZId (Proposition 4.2). 2. Preliminaries 2.1. Rings and f-rings. All rings (including f-rings) in this paper are commutative with identity 1. We recall that an f-ring is a lattice-ordered ring A which satisfies the condition that, for any a; b 2 A and any c ≥ 0 in A, (a ^ b)c = (ac) ^ (bc): By a positive element of an f-ring A we mean an element a ≥ 0, and we set A+ = fa 2 A j a ≥ 0g: Squares are positive in f-rings. An f-ring has bounded inversion if every element a ≥ 1 is invertible. A ring is semiprime if it has no nonzero nilpotent element. By the Jacobson radical of a ring we mean the intersection of all its maximal ideals. We write Max(A) for the set of all maximal ideals of a ring A. For any a 2 A, we set M(a) = fM 2 Max(A) j a 2 Mg: An ideal I of A is a z-ideal if, for any a; b 2 A, M(a) = M(b) and a 2 I imply b 2 I. See [11] for a detailed study of z-ideals in rings. Every maximal ideal is a z-ideal, and every minimal prime ideal is a z-ideal. Intersections of z-ideals are z-ideals, and the Jacobson radical of A, here denoted Jac(A), is a z-ideal contained in every z-ideal. An ideal of C(X) is a z-ideal in the usual sense if and only if it is a z-ideal in the algebraic sense as just defined. 2.2. Algebraic frames. The lattices we shall deal with are of a special kind that includes lattices of open sets of a topological space. They are called frames. To recall, a frame is a complete lattice L such that the distributive law _ _ a ^ S = fa ^ x j x 2 Sg A NOTE ON LATTICES OF z-IDEALS OF f-RINGS 353 holds for every a 2 L and S ⊆ L. We denote by 0 the zero of a frame, and by 1 its unit. A frame homomorphism is a mapping h: L ! M between frames that preserves finite meets (including the unit), and arbitrary joins (includ- ing the zero). The resulting category is denoted by Frm. Our references for frames are [5] and [12]. We write k(A) for the set of compact elements of a frame A. If k(A) generates A, in the sense that every element of A is the join of compact elements below it, then A is said to be algebraic. An algebraic frame A is said to have the finite intersection property (FIP) if a^b 2 k(A) for all a; b 2 k(A). A compact algebraic frame with FIP is called coherent, as is a frame homomorphism φ: A ! B between coherent frames that takes compact elements to compact elements. As usual, CohFrm denotes the category of coherent frames with coherent frame homomorphisms. A coherent frame is called a Yosida frame [10] if every compact element in it is a meet of maximal elements. 3. Frames of z-ideals of f-rings As stated in the Introduction, we aim to show in this section that some of the properties of the lattices Cz(X) are mainly due to the fact that C(X) is an f-ring with bounded inversion. Recall that an ideal I of an f-ring A (or `-ring, for that matter) is an `-ideal if, for any a; b 2 A, jaj ≤ jbj and b 2 I =) a 2 I: As remarked in [7], if A is an f-ring with bounded inversion, then every z- ideal in A is an `-ideal. In fact, an f-ring A has bounded inversion precisely when every z-ideal in A is an `-ideal. Indeed, if every z-ideal is an `-ideal, then every maximal ideal is an `-ideal, which is known to be equivalent to having the bounded inversion property. We denote by ZId(A) the lattice of z-ideals of A. Banaschewski shows in [2, Proposition 2.4] that the lattice RL(A) of radical `-ideals of any f-ring A is a coherently normal frame. A frame A is coherently normal if it is coherent, and for each compact c in A, the frame #c is normal. We show that for any f-ring A with bounded inversion, ZId(A) is a quotient of RL(A). This we do by defining a nucleus z on RL(A) for which Fix(z) = ZId(A). For a 2 A, set \ M(a) = fM 2 Max(A) j a 2 Mg: The convention, as usual, is that if a belongs to no maximal ideal (which is the case precisely when a is invertible) then M(a) is the entire ring. Note that, being an intersection of z-ideals (recall that maximal ideals are z- ideals), M(a) is a z-ideal, and is, in fact, the smallest z-ideal containing a. It is clear that M(a2) = M(a) since our maximal ideals are prime. Also, M(0) = Jac(A). Observe that, in any ring A, if I ⊆ A is an ideal, then I is a z-ideal () 8a 2 I;M(a) ⊆ I: 354 THEMBA DUBE We need the following lemma, which we state more generally that will be needed for our purposes. Lemma 3.1. Let A be an f-ring with bounded inversion, and let a; b 2 A. (a) M(a) \ M(b) = M(ab). (b) If a ≥ 0 and b ≥ 0, then M(ab) = M(a ^ b). (c) If a ≥ 0 and b ≥ 0, then M(a) + M(b) ⊆ M(a + b). Proof. (a) Let x 2 M(a) \ M(b). If N is a maximal ideal containing ab, then N contains a or it contains b, by primeness. In either case, x 2 N since x belongs to every maximal ideal containing a, and to every maximal ideal containing b. Thus, x 2 M(ab), showing that M(a)\M(b) ⊆ M(ab). On the other hand, ab belongs to every ideal that contains a, and every ideal that contains b. Therefore ab 2 M(a) \ M(b), and hence M(ab) ⊆ M(a) \ M(b) since M(a) \ M(b) is a z-ideal. Thus, M(ab) = M(a) \ M(b). (b) Since ab = (a_b)(a^b), ab 2 M(a^b), hence M(ab) ⊆ M(a^b). Also, 0 ≤ (a ^ b)2 ≤ ab 2 M(ab), which implies M(a ^ b) = M((a ^ b)2) ⊆ M(ab).
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